Characterization of Decentralized and Quotient Fixed Modes via Graph Theory

This paper deals with the decentralized control of systems with distinct modes. A simple graph-theoretic approach is first proposed to identify those modes of the system which cannot be moved by means of a linear time-invariant decentralized controller. To this end, the system is transformed into its Jordan state-space representation. Then, a matrix is computed, which has the same order as the transfer function matrix of the system. A bipartite graph is constructed from the computed matrix. Now, the problem of characterizing the decentralized fixed modes of the system reduces to verifying if this graph has a complete bipartite subgraph with a certain property. Analogously, a graph-theoretic method is presented to compute the modes of the system which are fixed with respect to any general (nonlinear and time-varying) decentralized controller. The proposed approaches are quite simpler than the existing ones, which often require calculating the rank of several matrices.